Question

Outpatient clinics are hospital clinics that have appointments for patients who visit the hospital without being admitted. At one particular clinic, it is known that the distribution of appointment times (in hours), X, is described by the cumulative distribution function:
??(?) = k?2 (?2 + 1), 0 ≤ ? ≤ 2.
a-   Find k such that FX(x) is a valid cdf
b-   What is the probability, Pr(? > 1)
c-   What is the probability, Pr (1/ 2 < X < 2)
d-   Find the expected value of X.
(show working)

Answer 1

(a)

Given:

,

                             0 x 2

By definition of the cumulative distribution function:

we have:
F(x=2) = 1.

Thus, we get:

Thus, we get:

So,

(b)

So,

the cumulative distribution function is given by:

                                  0 x 2

P(X>1) = 1 - F(X=1)

Substituting, we get:

P(X>1) = 1 - F(X=1) = 1 - 0.1 = 0.9

So,

Answer is:

0.9

(c)

P(1/2 < X < 2) = F(X=2) - F(X = 1/2)

Substituting, we get:

P(1/2 < X < 2) = F(X=2) - F(X = 1/2) = 1 - 0.015625 = 0.984375

So,

Answer is:

0.984375

(d)

Probability Density Function f(x) is got by differentiating F(x) as follows:

So,

we get:

So,

the expected value of X is given by:

between limits 0 to 2.

Applying limits, we gt:

E(x) = 1.5467

So,

Answer is:

1.5467